Nonlinear stability and transition threshold for the planar helical flow

In this paper, we study the nonlinear stability for the 3-D planar helical flow $(\delta^2\sin(m_0 y),\delta^2\cos(m_0 y),0)$ on torus $\mathbb{T}^3=\{(x_1,x_2,y)\big|x_1,x_2\in \mathbb{T}_{2\pi}, y\in \mathbb{T}_{2\pi \delta}, \delta\geq1\}$ for high Reynolds number $Re$. We prove that if the initial velocity $U_0$ satisfies $$ \left\|U_0-(\delta^2\sin(m_0 y),\delta^2\cos(m_0 y),0)\right\|_{X_0}\leq c_0 Re^{-7/4} $$ for some $c_0>0$ independent of $Re$, then the solution of 3-D incompressible Navier-Stokes equation is global in time and does not transit away from the planar helical flow. Here $\delta>1, m_0=\delta^{-1}$ and the norm $\|\cdot\|_{X_0}$ is defined in (1.8). This is a nonlinear stability result for 3-D non-shear flow and the transition threshold is less than $7/4$.